Applying algorithm finding shortest path in the multiple- weighted graphs to find maximal flow in extended linear multicomodity multicost network

The shortest path finding algorithm is used in many problems on graphs and networks. This article will introduce the algorithm to find the shortest path between two vertices on the extended graph. Next, the algorithm finds the shortest path between the pairs of vertices on the extended graph with multiple weights is developed. Then, the shortest path finding algorithms is used to find the maximum flow on the multicommodity multicost extended network is developed in the article [12]. Key word: Graph; Network; Multicommodity Multicost flow; Optimization; Linear Programming. Received on 12 October 2017, accepted on 7 December 2017, published on 21 December 2017 Copyright © 2017 Chien Tran Quoc and Hung Ho Van et al., licensed to EAI. This is an open access article distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unlimited use, distribution and reproduction in any medium so long as the original work is properly cited. doi: 10.4108/eai.21-12-2017.153499 ________________________ 2 Corresponding author: Hung Ho Van. Email: hovanhung@qnamuni.edu.vn


Introduction
The shortest path finding algorithm is used in many problems on graphs and networks. This article will introduce the algorithm to find the shortest path between two vertices on the extended graph. Next, the algorithm finds the shortest path between the pairs of vertices on the extended graph with multiple weights is developed. Then, the shortest path finding algorithms is used to find the maximum flow on the multicommodity multicost extended network is developed in the article [12].

The problem of finding the shortest path in extended graph
Given extended graph G = (V, E) with a set of vertices V and a set of edges E, where edges can be directed or undirected. Each edge eE is assigned a weight we(e). For each vertex vV, we denote Ev the set of edges incident vertex v. For each vertex vV and each of pair of edges (e,e')EvEv, ee' is assigned switch weight wv(v,e,e').
The sets (V, E, we, wv) are called extended graph.
Let p be a path from a vertex u to a vertex v through the edges ei, i = 1, …, h+1, and vertices ui, i = 1, …, h, as following: e1, u1, e2, u2, …, eh, uh ,eh+1, v] (1) Define the length of the path p, denoted l(p), as following: • The problem of finding the shortest path Given extended graph G = (V, E, we, wv) and vertices s, tV . Find the shortest path from s to t.
• Algorithm ◊ Input. The extended graph G = (V, E, we, wv) and vertices s, tV. Assign to l(t)=L(t,le(t)); // shortest path length from s to t.
// Moves from t, in reverse direction, to the preceding vertex-edges, we get the shortest path as follows: The algorithm that finds the shortest path in the extended graph is correct and has an algorithmic complexity of O(n 3 ) (n is the number of vertices in the graph).

The problem of finding the shortest path on the multiple-weighted extended graph
Given extended graph G = (V, E) with a set of vertices V and a set of edges E, where edges can be directed or undriected. On the graph there are r edge weights wei and switch weights wvi, i=1..r.
Let p be the path from the u to v through the edges ei, i = 1, …, h+1, and vertices ui, i = 1, …, h, as follows Define the length of the path p by edge weight wei and switch weights wvi, the symbol li(p), i=1..r, using the following formula:  Applying algorithm finding shortest path in the multiple-weighted graphs to find maximal flow in extended linear multicomodity multicost network 3 The problem is to find, among the source-destination pairs (si,j, ti,j), i=1..r, j=1..ki, the one that has the smallest shortest path length.
The source-destination pairs (si,j, ti,j), i=1..r, j=1..ki. ◊ Output. The source-destination pair (simin,jmin, timin,jmin) with the smallest shortest path length. lmin is the shortest path length from simin,jmin to timin,jmin, and the shortest parth Proof. The correctness of the algorithm derives from theorem 2.1. The algorithm that finds the shortest path between the source-destination vertices has the complexity O(n 3 ), which inferred the algorithm finding the smallest shortest path between the k of the destination source pair has complexity O(k.n 3 ).

The problem of maximum flow on extended Linear multicommodity multicost network
The model of multicommodity multicost network was built in the article [12].
Given a multicommodity multicost extended G=(V,E, ce, ze, cv, zv, {bei, bvi, qi |i=1..r}). Assume for each commodity i, i=1..r, with ki source-destination pairs (si,j, ti,j), j=1..ki, each of pair assigned a quantity of commodity type i, which needs to be transferred from source node si,j to destination node ti,j.
The problem is to find the multicommodity flow such that the flow value is maximal.
for e E : le(e)=; xi,j(e)=0 ; Using the algorithm to find the source-destination pair (si,j, ti,j), 1ir and 1jki, with the smallest shortest path from si,j to ti,j with edge weight le(e), eE, and swicth weights at nodes are lv(v), vV.
Symbol imin và jmin are index pairs of the source-destination nodes has the shortest path.  where n is the number of vertices, m is the number of edges and k=k1+…+kr. Proof. See [12].

Example
Showing an extended network diagram in Figure 1. The data given in the following tables Applying algorithm finding shortest path in the multiple-weighted graphs to find maximal flow in extended linear multicomodity multicost network 5

Conclusion
The article develops the algorithm finding the shortest path in extended graphs (Section 2), the algorithm finding the shortest path on the multiple-weighted extended graph (Section 3). Based on the duality theory of linear programming, an approximation algorithm with polynomial complexity is developed on the base of the algorithm finding shortest paths in section 2 and 3. This is also the main result of the article. Correctness and algorithm complexity are justified and the algorithm is stored in C++ and given an exact result. The results of this article are the basis for studying the applications of multicomodity multicost flow optimization .